Linear Optimization over a Polymatroid with Side Constraints – Scheduling Queues and Minimizing Submodular Functions

Two seemingly unrelated problems, scheduling a multiclass queueing system and minimizing a submodular function, share a rather deep connection via the polymatroid that is characterized by a submodular set function on the one hand and represents the performance polytope of the queueing system on the other hand. We first develop what we call a grouping algorithm that solves the queueing scheduling problem under side constraints, with a computational effort of O(n), n being the number of job classes. The algorithm organizes the job classes into groups, and identifies the optimal policy to be a priority rule across the groups and a randomized rule within each group (to enforce the side constraints). We then apply the grouping algorithm to the submodular function minimization, mapping the latter to a queueing scheduling problem with side constraints. We show the minimizing subset can be identified by applying the grouping algorithm n times. Hence, this results in a fully combinatorial algorithm that minimizes a submodular function with an effort of O(n).